bcorsten-journal-manager-256-1012-1-ce

Shipment Consolidation Ellissa Verseput ∗ Abstract In this paper, methods to realize cost savings for Airbus Helicopters France are considered This company makes a lot of shipments on different origin[.]

Shipment Consolidation

Ellissa Verseput ∗

Abstract

In this paper, methods to realize cost savings for Airbus Helicopters France are considered This company makes a lot of shipments on different origin-destination pairs By consolidating different shipments, bigger shipments can be sent to which lower cost/kg apply In order to schedule such consolidations when orders come in

at short notice, different policies are considered and a mixed integer linear program is designed These methods are implemented and tested on different test instances Their performances on cost savings and running times are compared and this shows that the policies would be more practical for day-to-day use, but that the mixed integer linear program generates more cost savings when Airbus Helicopters France strictly has to meet its customer’s preferences

Airbus Helicopters France (AH-F) has to deal with a lot of shipments to and from their customers At the moment the planning of these shipments seems far from optimal and the shipments are not kept track of and planned in an organized way AH-F has hired the DHL LLP team, a supply chain consulting team within DHL, to bring structure to the different shipments and make optimization recommendations for their shipments planning

in the future

AH-F makes use of different carriers to do their shipments, among them DHL itself AH-F operates from four warehouses, in Marseille, Paris and Hong Kong, from and to which

∗ Ellissa Verseput received a bachelor degree in Econometrics & Operations Research at Maastricht Uni-versity in 2015, where she currently takes the Master in the same field.

Contact: e.verseput@student.maastrichtuniversity.nl

export and import takes place AH-F sends shipments both by road and by air For each shipment the client currently determines the service type The choice is between a quick and customer-friendly, the so called ”aircraft on ground” (AOG), service and a standard routine service The service type chosen determines the time the shipment takes to arrive on its destination (the lead time)

DHL LLP retrieved data about shipments that have been made during 2014 As often in practice, the data is very incomplete, which makes analysis more complicated Nevertheless each shipment should in theory have a couple of properties A shipment in the past data has an origin airport and a destination airport Moreover, for each shipment the carrier that executed the shipment is known, as well as the shipment type (road or air), the service type (AOG or routine), weight of the parcel, pick up date, lead time and total paid costs

The information given by the past data needs to be translated to a planning system that can be used in the future So, knowing or expecting that certain shipments need to be made

in the future, a framework needs to be created that can schedule these shipments in a better way than is done at the moment Part of AH-F’s business exists of orders that arrive at short notice before their deadlines This makes it interesting to look into models and policies that work with orders that arrive at short notice Such models and policies can potentially

be tailored to the AH-F case in order to achieve cost savings This is the problem which this paper will deal with

The method for achieving cost savings in this paper is shipment consolidation This means that individual shipments of the same origin-destination pair will be combined in order to achieve cost savings, as the combined total weight will fall in a lower cost/kg rate Routing will be treated as given, assuming that only specific airport-to-airport origin-destination pairs are considered for consolidation Hence, consolidation will be optimized per unique origin-destination pair Also it will be assumed that the cheapest carrier and transport mode can be chosen in the future, so that one cost rate table always applies to a certain origin-destination pair Strictly speaking, inventory costs also should be taken into account when consolidation is done, because orders will need to be stored for a longer period

of time before they are sent together with other orders However, AH-F did not provide any information about their inventory structure, so it is not considered for this paper

In the following, first, an overview of past work on optimization models and policies of freight transportation with a focus on consolidation will be presented in Section 2 Then in Section 3, the consolidation problem that will be worked with in this paper is formulated, after which different consolidation policies for this problem will be introduced in Section 4

A mixed integer linear program formulation of the consolidation problem will be described

in Section 5 Test instances will be created in Section 6 and their results will be presented

in Section 7

2 Literature review

Freight transportation optimization is an area of operations research and supply chain man-agement that has been researched a lot in the past decades Many companies nowadays have a business aspect that involves freight transportation High efficiency and great service quality are the standard So, improving freight transportation and finding new ways of cost savings are booming business Crainic and Laporte (1997) give a very broad overview of all the different levels at which freight transportation can be optimized Strategic planning, the long term planning horizon, involves issues like network design and the locating of facilities

In tactical planning, the medium term planning horizon, service network design and vehicle routing problems are at hand For the short term operational planning, scheduling and the allocation of resources needs to be decided on In this paper, optimizing the tactical and operational planning are considered, as consolidation policies are determined on the tactical level and the actual scheduling of shipments is done at the operational level

Higginson and Bookbinder (1994) describe and analyze three consolidation policies that are used for combining shipments The time policy consolidates until the first order that came in has reached a certain age The quantity policy combines shipments until a certain quantity is accumulated A time and quantity policy does the previous at the same time; whichever constraint (time or quantity) is binding first, determines when a consolidation

is released By simulation and trying different parameters, they draw conclusions in which case which policy performs best Bookbinder and Higginson (2002) compare consolidation

of stochastically incoming orders having a Poisson distribution with a general stochastic clearing system in order to obtain the maximum holding time and maximum quantity

C¸ etinkaya and Bookbinder (2003) analytically derive optimal parameters for the time and quantity policy, again using that orders come in with a Poisson arrival process Mutlu

et al (2010) also analytically derive parameters for a hybrid time and quantity policy Both papers rely heavenly on stochastic properties given by the Poisson distribution

Instead of applying the standard consolidation policies, Tyan et al (2003) formulate an integer optimization program for different service policies, applied to a real life case They solve the integer program by using integer optimization software (Lingo), which is possible because the instances are not very big Attanasio et al (2007) do a case study as well, where besides consolidation constraints, also bin packing constraints are at hand They solve the problem by first solving an (infeasible) integer linear program and then iteratively making the simplified solution feasible Song et al (2008) also formulate an optimization problem for

a specific consolidation problem and recognize that it is NP hard They design a heuristic to solve the optimization problem and compare it to solution retrieved (very slowly) by CPLEX Qin et al (2014) do a case study on consolidation by designing a heuristic for a variant that takes different containers and different routes in account besides consolidation

3 Problem formulation

Following the literature, consolidation policies can offer AH-F a practical and easy to imple-ment framework for the future Solving their consolidation problem with an exact approach, like a mixed integer linear program (MILP) is also an option, although in practice it will probably be harder to implement for day-to-day use This paper will both look into consoli-dation policies and an MILP solution and compare their results The policies and the MILP will be applied to the following problem description:

• We consider one origin-destination pair

• At the beginning of every day, all new orders of that day come in together

• Those orders always need a fixed number of days before they are ready for shipping

• From the moment that orders are ready for shipping, they have (heterogeneous) dead-lines for arriving on destination that have to be met

• Every order has a weight

• The customer has chosen a service type (AOG or routine), but AH-F could decide to neglect this chosen service type, and see which cost savings this could bring, especially

to have some more freedom to build consolidations for the policies

• The AOG and routine service both have a fixed lead time

• The orders need to be sent as cheap as possible, but without violating their deadlines (and service type in case we consider this as something that cannot be neglected)

The transportation costs of the AOG service are higher than the routine service The transportation cost function, a function that only has the total weight of the shipment as an input, is non-decreasing and piecewise linear, such that the cost/kg rates are non-increasing There will be no limit opposed on the total weight of a shipment For visualization, an abstract transportation cost function f (w) description and a sketch look as follows:

f (w) =











M AX(P0; w ∗ p0) if q0 < w ≤ q1

M AX(P1; w ∗ p1) if q1 < w ≤ q2

M AX(Pn; w ∗ pn) if qn< w

(3.1)

with q0 = 0, p0 > p1 > > pn, P0 < P1 < < Pn, qj+1∗ pj = Pj+1

Figure 3.1: Sketch of f (w)

As described in section 2, Higginson and Bookbinder (1994) already pointed out different policies that can be used to consolidate shipments These policies can potentially provide more structure and efficiency for AH-F Take in mind that these policies should offer an easy to implement schedule advice for day-to-day use So the policies described certainly do not lead to the most optimal schedule possible, but are just structural guidelines in order to achieve a decent consolidation scheme Also notice that the fixed days between the day a shipment becomes known and is ready for shipping does not influence the policies described below

In the time policy described by Higginson and Bookbinder (1994), consolidation is done

on the basis of the oldest order approach; As soon as the oldest order that is waiting to be shipped has reached a certain predetermined age T (in days), all the orders that are currently available to be shipped are consolidated and sent However, the models in which a fixed T is used do not deal with heterogeneous deadlines Hence, the time policy can not immediately

be applied as such to AH-F, which also means that the analytical results of C¸ etinkaya and Bookbinder (2003) do not apply for determining the optimal T for a consolidation Instead, tailor made time policies will be tested on the test instances Given the lead times of the

routine service type and the AOG service type, the deadline of an order that becomes binding first can be used to determine the time when a consolidation is shipped Three policies are formulated:

AOG time policy In this policy, AH-F will choose the service type used and the AOG lead time is used to determine when a deadline becomes binding So whenever the AOG lead time until the deadline of a certain order becomes the exact time left till the deadline, all orders that are currently waiting to be shipped are consolidated and sent Under this policy, there is often more time to wait before shipping, as the AOG service is relatively faster However, the AOG service is more expensive than the routine service Nevertheless,

as larger consolidations could be build up when shipping is delayed, the total weight will fall into a lower cost/kg rate, which could be cheaper than shipping smaller consolidations with the routine service

Routine time policy In this policy the routine lead time determines when a deadline

of a certain order becomes binding This policy can profit from the lower cost rates of the routine service, while there is in general less time to build up a consolidation before a deadline becomes binding If an order comes in which lead time is so short that it will not meet the deadline when it is shipped by the routine service, it will have to be shipped by the more expensive AOG service As soon as this AOG lead time becomes binding, a couple

of different sub-policies can be implemented:

• Routine time policy A: the urgent order that needs to be shipped by AOG is shipped separately So all orders that are still feasibly shipped with the routine service are shipped in consolidation with the routine service Although the AOG shipment will

be relatively expensive, the routine consolidation is not touched and can hopefully still

be sent with a lower cost/kg rate because of large weight and the lower routine service rates

• Routine time policy B: all orders that are currently waiting are being shipped in con-solidation with the urgent shipment with the AOG service In this case, there is at least the potential for the AOG consolidation to fall into a lower cost/kg rate

Customer chooses time policy Because the assumption that AH-F can neglect the service type chosen by the customer is quite strong, this policy will meet the customer’s preferences (as the AOG service is not only faster, but it also comes with extra other services)

So AOG and routine orders will have to be consolidated separately Again, as soon as an AOG deadline will become binding, an AOG consolidation is sent with the current AOG orders and when a routine deadline will become binding a routine consolidation is sent

Immediate shipping For comparing the above explained policies, the standard policy

in which the incoming orders are just shipped right away with the preferred service type

is implemented as well In this policy, all orders that have come in on a certain day with preselected AOG service are immediately sent together and the same goes for the routine service Notice that orders that have come in on the same day are still sent in consolidation,

as the orders arrived together as well

4.2 Quantity policy

The quantity policy described by Higginson and Bookbinder (1994), that sends out an con-solidation whenever a certain target weight W has been accumulated, is not applicable to AH-F, as it cannot guarantee that the deadlines are met Hence, implementing some kind

of quantity policy does not make any sense

The policies described in the foregoing section can provide cost savings already, but are not very sophisticated They are just logical rules of thumb Especially when there is no flexibility in neglecting the chosen service type, the policies do not have a lot of freedom

to make consolidations So in that case, something more sophisticated might be needed to provide substantial cost savings The following MILP solution of the consolidation problem will take more computation time, but provides the optimal consolidation schedule seen from today’s perspective See Appendix A for a list of the used parameters and variables

Considering a discrete rolling time horizon, the model will be optimized every day from the perspective of the current day ¯t, when new orders have come in Remember it is assumed that at the beginning of each day t, all new orders that come in that day are known At that moment, the scheduling and the consolidation of the current known orders is optimized and the orders that need to be sent today according to the current solution are then sent Let I¯t be the set of orders that are known on the current day ¯t Each i ∈ I¯t has a couple

of attributes The arrival day: ai, the number of days until the deadline after the order is ready for shipping: di, weight of the order: wi and binary parameters for the service type selected by the customer: ki,s, s ∈ {AOG, routine} The lead times for the two service types, AOG and routine, are indicated with the parameters ls, s ∈ {AOG, routine} The fixed time between the arrival of an order and the day that the order is ready for shipping

is the parameter r

Every day, we need to plan from today ¯t until day T , where T = maxi{ai + r + di}

In order to assign each order i ∈ It to a consolidation, binary decision variables xi,t,s are introduced, which equal 1 when order i is sent on day t = ¯t, , T with service type s ∈

{AOG, routine} Because of the non-increasing cost/kg structure of the transportation cost described in equation (3.1), all orders that are sent on a certain day with the same service type, are best sent in one consolidation, as this will always be cheaper per kg than sending

it with separate orders That is why the decision variables xi,t,s are sufficient to determine the consolidations that are sent today and in the upcoming days

The transportation cost function f of a consolidation of orders can be calculated using the total accumulated weight that is sent by it So f ’s input is P

i∈I ¯ t(xi,t,s∗ wi)

In principal the following integer program needs to be solved on current day ¯t:

Min X

s

T

X

t=¯ t

f (X

i∈I ¯ t

s.t X

s

T

X

t=¯ t

X

s

T

X

t=¯ t

(t ∗ xi,t,s) ≥ ai+ r, ∀i ∈ I¯t (5.3)

X

s

T

X

t=¯ t

((t + ls) ∗ xi,t,s) ≤ ai+ r + di, ∀i ∈ I¯t (5.4)

T

X

t=¯ t

(5.2) ensures that each order is assigned to only one consolidation (5.3) makes sure that the orders are not sent before they are available (5.4) takes care that the orders arrive before their deadlines (5.5) ensures that the orders are sent by the correct service type However, the objective function (5.1) is not linear, so the integer program can not be solved like this We need to add extra binary variables and modify the objective function

in order to make this an integer linear program Namely, problems with piece-wise linear objective functions like this can be formulated as mixed integer linear programs

Before we do that, we first slightly change the presentation of the transportation cost function f Remember in equation (3.1) we had:

f (w) =









M AX(P0; w ∗ p0) if q0 < w ≤ q1

M AX(P1; w ∗ p1) if q1 < w ≤ q2

M AX(Pn; w ∗ pn) if qn< w with q0 = 0, p0 > p1 > > pn, P0 < P1 < < Pn, qj+1∗ pj = Pj+1

Figure 5.1: Sketch of new presentation of f (w)

Now consider the m fixed breakpoints of this function b1, , bm and their fixed function value f (b1), , f (bm), where bm = B, with B chosen sufficiently large, such that no total weight of any consolidation will exceed B (see figure 5.1) Introducing additional real-valued variables yj ∈ [0, 1], j = 1, , m, we can rewrite f (w) as follows:

f (w) =

m

X

j=1

yj ∗ f (bj) (5.7)

m

X

j=1

yj ∗ bj = w (5.8)

m

X

j=1

(5.8) writes the original weight as a linear combination of the breakpoint weights and this makes sure that the cost for the original weight is calculated as a linear combination of the cost of the breakpoint weights in (5.9) (5.10) ensures that the yjs really make a linear combination by making them add up to 1 Now only one more thing is needed, namely that only two consecutive yjs are larger than 0 Otherwise, the linear combination will not represent a point on the original cost function line In order to do this, we need the yjs to

be so called ”Special Ordered Set of type 2” variables (SOS2 variables), which simply means that only two consecutive yjs are larger than 0 in an ordered set of yjs The additional mathematical constraints and binary variables zjs that ensure this are:

m−1

X

j=1

yj ≤ zj−1+ zj, j = 1, , m (5.11)

zj ∈ {0, 1} , z0 = zm = 0, yj ∈ [0, 1] (5.12) (5.10) imposes that only one zj can be equal to 1, say for j = ¯j Hence (5.11) makes sure that only y¯j and y¯j+1 can be larger than 0 These techniques have been retrieved from ideas explained by Bisschop (2006) and DAmbrosio (2010) Having introduced this new presentation of f (w), we are ready to formulate the MILP again: